Quaternions, Rotations and Real numbers - Mathematics Stack …
Dec 11, 2017 · Then a general quaternion (a constant times a unit quaternion) can be thought of as representing a linear transformation which is a combination of a rotation and a scaling, and the multiplication of two quaternions is an algebraic way of computing the composition of two such linear transformations: $$ q_1 \bigl( q_2 u q_2^* \bigr) q_1^* = (q_1 ...
Real world uses of Quaternions? - Mathematics Stack Exchange
The quaternion algebra shows there as a way of disentangling two Alamouti coded signals transmitted by a pair of antennas. The advantages come from the fact that even if the signal from one antenna is lost for a particular receiver (due to sitting in a node for that particular radio wave), then the signal from the other antenna saves the day.
How can one intuitively think about quaternions?
Oct 19, 2010 · Here is the intuitive interpretation of this. Given a particular rotation axis $\omega$, if you restrict the 4D quaternion space to the 2D plane containing $(1,0,0,0)$ and $(0,\omega_x,\omega_y,\omega_z)$, the unit quaternions representing all possible rotations about the axis $\vec \omega$ form the unit circle in that plane.
matrices - Quaternion - Conjugate - Mathematics Stack Exchange
May 4, 2018 · But if I apply this to the Quaternion I have it looks as if it is just subtracting each ijk from each other, or maybe I'm just not understanding this correctly. I'd like to better understand what is going on here and how to get the conjugate of a Quaternion properly. I'm happy to follow any web links that better explain the method of doing this.
How do quaternions represent rotations? - Mathematics Stack …
In the quaternion case, reduced means that instead of taking this as the norm, you take its square root. Since the quaternions are 4-dimensional over $\Bbb R$, the reduced norm defines a quadratic form, which is what one would expect from an euclidean norm. $\endgroup$
2 四元数变换(Quaternion Transforms) - 知乎
推导过程参考：四元数和旋转(Quaternion & rotation)，内容较为详细。 一旦构建好四元数，就不需要计算三角函数了。因此实际上这个转换过程是高效的。 把一个正交矩阵 转换为一个四元数 会稍微复杂一点。关键点是下面从方程4.46中得到的差: 而矩阵 的迹为:
Concise description of why rotation quaternions use half the angle
Aug 5, 2015 · Every quaternion multiplication does a rotation on two different complex planes. When you multiply by a quaternion, the vector part is the axis of 3D rotation. The part you want for 3D rotation. But you ALSO do a rotation in the complex plane consisting of …
Quaternions and spatial translations - Mathematics Stack Exchange
$\begingroup$ The alternative is the dual quaternion, which Gerard mentioned (albeit not by name) - they are composed the same way, the "sandwich" (a)(bcb*)(a*) = (ab)c(ba) = (ab)c(ab)*, which means a long sequence on the LHS only needs to be conjugated after the fact to find the RHS. So, you don't need to break a long sequence of quaternions ...
Quaternion , DCM , Euler Angles and Rotation Matrix Differences …
Quaternion, DCM[Direction Cosine Matrix] , Euler Angles and Rotation Matrix Differences and when to use ? All of the above components can represent rotation , so when to use each of them , best approach and pros ,cons of each . I've studied all of them but I feel I'm lost some how .
四元数和旋转(Quaternion & rotation) - 知乎
四元数(quaternion)可以看作中学时学的复数的扩充，它有三个虚部。形式如下： ，可以写成 . 具有如下性质： 设 ， ，则. 3.2 共 轭四元数. 一个四元数 的共轭(用 表示)为 . 一个四元数和它的共轭的积等于该四元数与自身的点乘，也等于该四元数长度的平方。即，